Product Rule of Differentiation

Q.

If ${\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z=\frac{\mathrm{\pi }}{2}$, then the value of $x^2+y^2+z^2+2xyz=$

1. $0$

2. $1$

3. $2$

4. $3$

Q.

Evaluate ${}^6{C}_3$

Q.

What is the derivative of $2x$?

Q.

What is the derivative of $co{t}^{-1}\left(x\right)$?

Q.

If ${\left[\frac{1+i}{1-i}\right]}^{\frac{m}{2}}={\left[\frac{1+i}{i-1}\right]}^{\frac{n}{3}}=1$, $(m,n\in N)$ then the greatest common divisor of the least values of $m$ and $n$ is

Q.

Find the value of ${\left(243\right)}^{\frac{1}{5}}$

Q.

What is the derivative of $5^x$?

Q.

If $f\left(x\right)=e^x,g\left(x\right)={\sin }^{-1}x$ and $h\left(x\right)=f\left(g\left(x\right)\right)$, then $\frac{h\left(x\right)}{h\left(x\right)}$ is equal to.

1. $e^{{\sin }^{-1}x}$

2. $\frac{1}{\sqrt{1-x^2}}$

3. ${\sin }^{-1}x$

4. $\frac{1}{1-x^2}$

Q. Differentiate the following equation:
$e^{x}log\sqrt{x}tanx$

Q. The locus of the foot of the perpendicular drawn from the center upon any tangent to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ is

1. $(x^2+y^2{)}^2=9x^2+16y^2$
2. $x^2+y^2=25$
3. $x^2+y^2=144$
4. $(x^2+y^2{)}^2=16x^2+9y^2$

Q.

Why is derivative of constant zero?

Q.

$2^{\frac{1}{4}}·4^{\frac{1}{8}}·8^{\frac{1}{16}}·{16}^{\frac{1}{32}}·.........$is equal to

1. $1$

2. $2$

3. $\frac{3}{2}$

4. $\frac{5}{2}$

Q.

If $3^x=4^{x-1}$, then $x$ is equal to

1. $\frac{2{\log }_{3}2}{2{\log }_{3}2–1}$

2. $\frac{2}{2-{\log }_{2}3}$

3. $\frac{1}{1-{\log }_{4}3}$

4. $\frac{2{\log }_{2}3}{2{\log }_{2}3–1}$

Q.

In any $∆ABC$, the simplified form of $\left(\frac{\cos 2A}{a^2}\right)-\left(\frac{\cos 2B}{b^2}\right)$ is:

1. $a^2-b^2$

2. $\frac{1}{a^2-b^2}$

3. $\frac{1}{a^2}-\frac{1}{b^2}$

4. $a^2+b^2$

Q.

If $p=-10$, Find the value of $p^2-2p-100$.

Q.

If ${\sin }^{3}x\sin 3x=\sum _{m=0}^n{C}_{m}cosmx$ where $c_0,c_1,c_2,\dots \dots \dots ..c_n$ are constants and $C_n\ne 0$, then the value of $n$ is:

1. $2$

2. $4$

3. $6$

4. $8$

Q.

If $k=\sin \frac{\pi }{18}\sin \frac{5\pi }{18}\sin \frac{7\pi }{18}$, then the numerical value of $k$ is

1. $\frac{1}{4}$

2. $\frac{1}{8}$

3. $\frac{1}{16}$

4. None of these

Q.

Evaluate $\tan \left[2{\tan }^{-1}\left(\frac{1}{5}\right)-\left(\frac{\mathrm{\pi }}{4}\right)\right]$

1. $\frac{17}{7}$

2. $-\frac{17}{7}$

3. $\frac{7}{17}$

4. $-\frac{7}{17}$

Q.

What does It mean when $\frac{dy}{dx}=0$?

Q.

Find the value of ${\left(64\right)}^{\frac{-2}{3}}$

Q.

$\sin ({\cot }^{-1}x)=$

1. $\sqrt{1+x^2}$

2. $x$

3. ${\left(1+x^2\right)}^{-\frac{3}{2}}$

4. ${\left(1+x^2\right)}^{-\frac{1}{2}}$

Q.

What is the derivative of the sum of two functions?

Q.

Differentiate the given functions w.r.t. x.

$(logx{)}^x+x^{logx}$

Q.

The value of $cot\left(\sum _{n=1}^{23}co{t}^{-1}\left(1+\sum _{k=1}^{n}2k\right)\right)$ is

1. $\frac{23}{25}$

2. $\frac{25}{23}$

3. $\frac{23}{24}$

4. $\frac{24}{23}$

Q.

The value of $\tan \left[2{\tan }^{-1}\left(\frac{3}{5}\right)+{\sin }^{-1}\left(\frac{5}{13}\right)\right]$ is

1. $\frac{220}{21}$

2. $\frac{110}{21}$

3. $\frac{55}{21}$

4. $\frac{20}{11}$

Q.

Use the identity$(x+a)(x+b)=x^2+(a+b)x+ab$ to find the following product.

$(4x-5)(4x-1)$

Q.

The rate of change of $\sqrt{(x^2+16)}$ with respect to $\left[\frac{x}{x-1}\right]$ at$x=3$is

1. $2$

2. $\frac{11}{5}$

3. $-\frac{12}{5}$

4. $–3$

Q.

The derivative of $e^{x^3}$ with respect to$\log x$ is

1. $e^{x^3}3x^3$

2. $3x^2{e}^{x^3}$

3. $e^{x^3}$

4. $3x^2{e}^{z3}+3x^2$

Q.

If $a,b,c\ne 0$ and belong to the set $\{0,1,2,3...9\}$, then ${\log }_{10}\left[\frac{(a+10b+{10}^{2}c)}{({10}^{-4}a+{10}^{-3}b+{10}^{-2}c)}\right]$

1. $1$

2. $2$

3. $3$

4. $4$

Q. Differentiate the following functions with respect to x:
sinx cosx